'65 Valiant No High Beam Problem

Just to be clear, it COULD HAVE been the switch, AKA something broken inside, shorting against the case.

HOW TO TROUBLESHOOT A BAD SHORT. I've preached and preached this, nothing on the OP, but here's a good way and it's always worked. Obtain for yourself a junk tail/ stop lamp socket and if possible, heavy duty bulb, AKA 1157. If possible, salvage any old headlamp that works with at least one good filament. Wire one of these rigs IN SERIES with the battery neg to ground. This will protect your wiring, because all that will happen when a heavy load is energized OR A SHORT occurs, is that the series bulb will light bright.

You can wire a tail/ stop socket a few ways to obtain different wattages.
A....heaviest. Wire both pigtail wires together, use that for one terminal, and the socket shell for the other
B....Lighter: Use only the turn / stop filament pigtail and the shell
C....Lighter yet: Use only the tail filament and the shell
D....Even lighter: Leave the shell unconnected and use the two pigtails for the two lamp terminals. This places the filaments in series.

So far as "what's left," think about what you have, and what it is.

1...A short (length) harness from the dimmer switch which ends up feeding the low and hi beam wires out through the bulkhead connector. Possible. Maybe the wiring where it runs under the carpet and up the side got damaged and shorted.

2...Hi and lo feeds go through bulkhead connector, unlikely

3..Underhood harness, runs from bulkhead connector, around side of fender, and out rad support to healight. Examine this closely, something caught on sharp metal, abraided, etc.

4..The lamps. It would be rare, but a lamp COULD SHORT internally. They get hot, the internal structure is weakened by the heat, and years of thermal cycling. You maybe hit a bump, and part of the internal structure sagged a bit. Easy. Unplug the lamps and see if problem goes away

5...Probably least likely but tuck this away in case. At some point in the wiring, the high beam feed branches off to feed up to the hi beam indicator lamp. It is possible that this wire shorted somehow.

I don't remember the bulkhead connector/ wiring in the old girls. It MIGHT be possible that you can disconnect the lighting harness under the hood separately, from the bulkhead, and still have other power. If the short goes away and the hi beam indicator works, you'll know the problem is in that harness or the lamps/.